This paper deals with distributed discrete-event systems, in which agents (or local sites) are requited to communicate in order to perform some specified tasks. Associated with each agent is a finite-state automaton that captures the required, tasks to be performed at that site. The problem considered is that each agent must be able to distinguish between the states of its automaton. To help it disambiguate states, an agent uses a combination of direct observation (obtained from sensor readings available to that agent) and communicated information (obtained from sensor readings available to another agent). Since communication may be costly a strategy to minimize communication between sites is developed. The complexity of the solution reflects the interdependence of the agents' communication protocols. That is, the decision to communicate the occurrence of an event relies on which event sequences are indistinguishable to an agent, which, in turn, is a result of what has already been communicated to that agent.